How can I make an exponentially curving object?
You can use the XYZ Math Surface available through the Extra Objects add-on to achieve this.

First, go into User Preferences, Add-ons, and search for Extra Objects. Enable it via the checkbox.



Now you should have additional objects in your Add/Mesh menu - in particular the Math Function, XYZ Math Surface. The XYZ Math Surface allows you to set up how two variables - U and V - vary, and how the X, Y, Z coordinates are generated from these.

Set up the parameters in the Toolshelf (T) as shown below :



Here U varies from 0 to 2*pi with 128 steps - one whole rotation (in radians).

V varies from 0 to 2.10 with 32 steps - this is the distance from the centre.

The X coordinate is defined as (v+0.1)*cos(u) + (v+0.1)*sin(u) and Y as -(v+0.1)*sin(u)+(v+0.1)*cos(u) - this converts from the polar coordinates to X and Y.

The Z is calculated using an exponential function -1/e**-(1.6-v) (EDIT - this can be simplified to e**(1.6-v)).

Note the +0.1 in each of the X and Y equations to offset the centre of the cone so we don't hit infinity (limits the inner edge). Also, adjusting the 1.6 in the Z equation adjusts the speed of the exponential falloff.